Universal Unitarity Triangle 2016 and the Tension Between $\Delta M_{s,d}$ and ${\epsilon }_K$ in CMFV Models

Abstract

Motivated by the recently improved results from the Fermilab Lattice and MILC Collaborations on the hadronic matrix elements entering $\Delta M_{s,d}$ in $B_{s,d}^0-{\bar{B}}_{s,d}^0$ mixing, we determine the Universal Unitarity Triangle (UUT) in models with Constrained Minimal Flavour Violation (CMFV). Of particular importance are the very precise determinations of the ratio $|V_{ub}|/|V_{cb}|=0.0864\pm 0.0025$ and of the angle $\gamma =(63.0\pm 2.1{)}^{\circ }$. They follow in this framework from the experimental values of $\Delta M_d/\Delta M_s$ and of the CP-asymmetry $S_{\psi K_S}$. As in CMFV models the new contributions to meson mixings can be described by a single flavour-universal variable $S(v)$, we next determine the CKM matrix elements $|V_{ts}|$, $|V_{td}|$, $|V_{cb}|$ and $|V_{ub}|$ as functions of $S(v)$ using the experimental value of $\Delta M_s$ as input. The lower bound on $S(v)$ in these models, derived by us in 2006, implies then upper bounds on these four CKM elements and on the CP-violating parameter ${\epsilon }_K$, which turns out to be significantly below its experimental value. This strategy avoids the use of tree-level determinations of $|V_{ub}|$ and $|V_{cb}|$ that are presently subject to considerable uncertainties. On the other hand if ${\epsilon }_K$ is used instead of $\Delta M_s$ as input, $\Delta M_{s,d}$ are found significantly above the data. In this manner we point out that the new lattice data have significantly sharpened the tension between $\Delta M_{s,d}$ and ${\epsilon }_K$ within the CMFV framework. This implies the presence of new physics contributions beyond this framework that are responsible for the breakdown of the flavour universality of the function $S(v)$. We also present the implications of these results for $K^{+}\to {\pi }^{+}\nu \bar{\nu }$, $K_L\to {\pi }^0\nu \bar{\nu }$ and $B_{s,d}\to {\mu }^{+}{\mu }^{-}$ within the Standard Model.

25 pages, 6 figures. v3: Numerical update in view of the modified results by the Fermilab-MILC collaboration with only minor impact on our results. References added

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